orderIsoSubobject — Mathlib

English

Range of a morphism between presheaves is a subpresheaf of the codomain.

Русский

Диапазон морфизма между префешами является подпрефешом кодомонада.

LaTeX

$$$ \\mathrm{range}(f) \\leq \\mathrm{Subpresheaf}(F') $$$

Lean4

/-- The order isomorphism `Subpresheaf F ≃o MonoOver F`. -/
@[simps]
noncomputable def orderIsoSubobject : Subpresheaf F ≃o Subobject F
    where
  toFun A := Subobject.mk A.ι
  invFun X := Subpresheaf.range X.arrow
  left_inv A := by simp
  right_inv X := by simp
  map_rel_iff'
    {A B} := by
    constructor
    · intro h
      have : range (Subobject.mk A.ι).arrow ≤ range (Subobject.mk B.ι).arrow :=
        leOfHom (((equivalenceMonoOver F).trans (ThinSkeleton.equivalence _).symm).inverse.map (homOfLE h))
      simpa using this
    · intro h
      exact leOfHom (((equivalenceMonoOver F).trans (ThinSkeleton.equivalence _).symm).functor.map (homOfLE h))

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